A263575 - OEIS

%I #17 Aug 06 2021 08:38:21

%S 2,1,4,14,53,227,1092,5791,33350,206511,1365563,9590847,71216713,

%T 556861216,4569168866,39222394456,351304769679,3275433717440,

%U 31723522878974,318571978752719,3311400814816987,35573458376435132,394404160256111139,4507130777468928696

%N Stirling transform of Lucas numbers (A000032).

%H Alois P. Heinz, <a href="/A263575/b263575.txt">Table of n, a(n) for n = 0..563</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F a(n) = Sum_{k=0..n} A000032(k)*Stirling2(n,k).

%F Let phi = (1+sqrt(5))/2.

%F a(n) = B_n(phi)+B_n(1-phi), where B_n(x) is n-th Bell polynomial.

%F 2*B_n(phi) = a(n) + A263576*sqrt(5).

%F E.g.f.: exp((exp(x)-1)*phi)+exp((exp(x)-1)*(1-phi)).

%F Sum_{k=0..n} a(k)*Stirling1(n,k) = A000032(n).

%F G.f.: Sum_{j>=0} Lucas(j)*x^j / Product_{k=1..j} (1 - k*x). - _Ilya Gutkovskiy_, Apr 06 2019

%t Table[Sum[LucasL[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]

%t Table[Simplify[BellB[n, GoldenRatio] + BellB[n, 1 - GoldenRatio]], {n, 0, 23}]

%Y Cf. A000032, A213593, A005248, A061084, A263576.

%K nonn

%O 0,1

%A _Vladimir Reshetnikov_, Oct 21 2015